The sum of two distinct positive integers is 25. The sum of the squares of these two integers is 337....

1. TRANSLATE the problem information

• Given information:
  • Two distinct positive integers have \(\mathrm{sum = 25}\)
  • The sum of their squares = 337
  • Need to find the smaller integer

• What this tells us:
  If the integers are x and y, then:
  • \(\mathrm{x + y = 25}\)
  • \(\mathrm{x^2 + y^2 = 337}\)

2. INFER the solution strategy

• We have two equations with two unknowns - a system of equations
• Since one equation is linear and the other is quadratic, substitution will be more efficient than elimination
• From the linear equation, we can express one variable in terms of the other

3. SIMPLIFY using substitution

• From \(\mathrm{x + y = 25}\), we get: \(\mathrm{y = 25 - x}\)
• Substitute into the second equation:
  \(\mathrm{x^2 + (25 - x)^2 = 337}\)

• Expand \(\mathrm{(25 - x)^2}\):
  \(\mathrm{x^2 + (625 - 50x + x^2) = 337}\)
  \(\mathrm{x^2 + 625 - 50x + x^2 = 337}\)
  \(\mathrm{2x^2 - 50x + 625 = 337}\)

4. SIMPLIFY to standard quadratic form

• Move all terms to one side:
  \(\mathrm{2x^2 - 50x + 625 - 337 = 0}\)
  \(\mathrm{2x^2 - 50x + 288 = 0}\)

• Divide everything by 2:
  \(\mathrm{x^2 - 25x + 144 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to 144 and add to -25
• Those numbers are -9 and -16: \(\mathrm{(-9) \times (-16) = 144}\) and \(\mathrm{(-9) + (-16) = -25}\)
• Factor: \(\mathrm{(x - 9)(x - 16) = 0}\)
• Solutions: \(\mathrm{x = 9}\) or \(\mathrm{x = 16}\)

6. INFER the corresponding y-values

• If \(\mathrm{x = 9}\), then \(\mathrm{y = 25 - 9 = 16}\)
• If \(\mathrm{x = 16}\), then \(\mathrm{y = 25 - 16 = 9}\)
• Both give us the same pair of integers: \(\mathrm{\{9, 16\}}\)

7. APPLY CONSTRAINTS to identify the final answer

• The problem asks for the smaller of the two integers
• Between 9 and 16, the smaller value is 9

Answer: B) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(25 - x)^2}\)

Many students incorrectly expand this as \(\mathrm{625 - x^2}\) instead of \(\mathrm{625 - 50x + x^2}\). This leads to the wrong equation:
\(\mathrm{x^2 + 625 - x^2 = 337}\), which simplifies to \(\mathrm{625 = 337}\) (impossible).

This leads to confusion and guessing among the answer choices.

Second Most Common Error Path:

Poor INFER reasoning: Students try to guess-and-check with the answer choices instead of setting up equations

Rather than creating a systematic algebraic approach, some students plug the answer choices into both conditions. While this can work, it's time-consuming and error-prone because they must check both the sum and sum-of-squares conditions for each potential pair.

This often causes them to get stuck and randomly select an answer when calculations become unwieldy.

The Bottom Line:

This problem tests whether students can work confidently with systems involving both linear and quadratic relationships. The key challenge is maintaining algebraic accuracy through the substitution and expansion process.